Cows and sheep
Use your logical reasoning to work out how many cows and how many
sheep there are in each field.
Age
7 to 11
Challenge level
Problem
Image
In this field we could say, because of the number of animals there, that each cow can see 4 sheep and 3 (other) cows. This could be worded as follows "Each cow can see one more sheep than cows." There are obviously 4 sheep and 4 cows in the field as you see it.
But here are some questions about different fields in which you have to find out how many sheep and cows there are in each field.
In field number 1, each cow can see twice as many sheep as cows; each sheep can see the same number of sheep as cows, so how many cows and sheep are there?
In field number 2, each cow can see three times as many sheep as cows; each sheep can see the same number of sheep as cows, so how many cows and sheep are there?
In field number 3, each cow can see twice as many sheep as cows; each sheep can see one more sheep than cows, so how many cows and sheep are there?
In field number 4, each cow can see twice as many sheep as cows; each sheep can see two more sheep than cows, so how many cows and sheep are there?
In field number 5, each cow can see three times as many sheep as cows; each sheep can see twice as many sheep as cows, so how many cows and sheep are there?
Getting Started
If there was just one cow, how many other cows would it see? How many sheep would the cow see? Does this fit the clues?
If there were two cows, how many would each cow see? How many sheep would each cow see? Does this fit the clues?
How about if there were three cows? Four cows ...?
Student Solutions
Before looking at some solutions that came in we would like to say that we were very pleased with the explanations that came in with many of the solutions - exactly what we would hope to encourage.
Francesca and Alex sent in the first correct solution and they come from Red Hill Field Primary School and so does Sam who sent in another correct solution a week later.
Francesca and Alex wrote:
Here is the solution:
In field no.1 there must be 3 cows and 4 sheep because cows
can't see themselves.
In field no.2 there must be 2 cows and 3 sheep.
In field no.3 there must be 4 cows and 6 sheep.
In field no. 4 there must be 5 cows and 8 sheep.
In field no. 5 there must be 4 cows and 9 sheep.
I know this because the animals cannot see themselves.
We had many good replies from Bradon Forest School and here's Samuel's correct solution with some good explanations as to how he went about it.
I worked out all of these questions by doing them step by
step...
1) Step 1: Firstly, you must take 1 cow and 2 sheep. This is
because each cow can see twice as many sheep than cows.
Step 2: It also says that each sheep can see the same number
of sheep as cows. This means that there must be a higher amount of
animals. Therefore there are 3 cows and 4 sheep . This is because
each cow can see 2 cows and 4 sheep (remember the cow can't see
itself) which is twice the amount of cows. and each sheep can see 3
cows and 3 sheep which is the same amount.
2) Step 1: First we take 3 times as many sheep as cows ( which
is 3 sheep and 1 cow).
Step 2: Next, it tells us that each sheep can see the same
number of cows and sheep. So therefore, we have to add one more
cow. This means that there are 2 cows and 3 sheep. This is because
each cow can now see one cow and 3 sheep and each sheep can see 2
sheep and 2 cows.
3) Step 1: Firstly, it says that each cow can see twice as
many sheep as cows. This means there are 2cows and 2 sheep.
Step 2: It then says that each sheep can see one more sheep
than cows, this means there are 6 sheep and 4 cows. This is because
each cow can see 3 cows and 6 sheep. This is twice the amount of
cows. It also means that each sheep can see 5 sheep and 4 cows.
There is one more sheep than cows.
4) Step 1: We are given the information that each cow can see
twice as many sheep than cows. This means there is 1 cow and 2
sheep.
Step 2: We are told that each sheep can see two more sheep
than cows. Therefore, there are 5 cows and 8 sheep. This means that
each cow can see 4 cows and 8 sheep which is twice as many sheep as
cows and each sheep can see two more sheep than cows ( 1 sheep can
see 7 sheep and 5 cows).
5) Step 1: We are told that each cow can see three times as
many sheep than cows. So let's start with 1 cow and 3 sheep.
Step 2: It says that each sheep can see twice as many sheep as
cows. This means there are 4 cows and 9 sheep. This is because 3
cows can see 9 sheep (3x3=9). Also each sheep can see 8 sheep and 4
cows which is twice the amount of cows.
Rees, who also came from Bradon Forest sent in a Powerpoint presentation and here's his slide for field 1
Image
Well done!
We had 71 replies come in about this problem, They came from far afield - Qatar, Dublin, North Carolina, Australia, New York, Wisconsin and New Zealand as well as the U.K. The problem caused some deep thinking and just 16 replies had the correct answer.
Teachers' Resources
Cows and Sheep
Image
In this field we could say, because of the number of animals there, that each cow can see 4 sheep and 3 (other) cows. This could be worded as follows "Each cow can see one more sheep than cows." There are obviously 4 sheep and 4 cows in the field as you see it.
But here are some questions about different fields in which you have to find out how many sheep and cows there are in each field.
In field number 1, each cow can see twice as many sheep as cows; each sheep can see the same number of sheep as cows, so how many cows and sheep are there?
In field number 2, each cow can see three times as many sheep as cows; each sheep can see the same number of sheep as cows, so how many cows and sheep are there?
In field number 3, each cow can see twice as many sheep as cows; each sheep can see one more sheep than cows, so how many cows and sheep are there?
In field number 4, each cow can see twice as many sheep as cows; each sheep can see two more sheep than cows, so how many cows and sheep are there?
In field number 5, each cow can see three times as many sheep as cows; each sheep can see twice as many sheep as cows, so how many cows and sheep are there?
Why do this problem?
This problem challenges children to think in an organised way - the main aim is to help them develop a system or logical approach for tackling it.
Possible approach
Begin by showing the group the picture in the problem. Rather than talking through the text underneath it, ask some questions about what they see, for example:
- What can you tell me about the number of sheep and cows in the picture?
- How many cows can each sheep see?
- How many cows can each cow see?
- How does the number of sheep a cow can see relate to the number of cows it can see?
Children might well say things like "each cow can see more sheep than cows", so ask them to be more exact. This may lead to statements like "each cow can see one more sheep than cows", or "each cow can see one less cow than sheep".
Try looking at the first field together. Simply read out what it says and ask the children to think about HOW they might find out the number of cows and sheep. Invite suggestions from the class. Choose a way to try which captures the idea of trial and improvement. If the children haven't had much experience of this, you might want to model a way for them as a start. So, focusing either on
the sheep's view or the cows' view, try the smallest number that there could be to start with, increasing this one at a time, checking to see whether the results fit with the information at each stage. Emphasise that this way of approaching the problem, (trying the simplest case and then working up a step at a time) is a very good way of going about solving it as it means no possibilities are
left out.
Key questions
What shall we try first?
If there was just one cow, how many sheep would it see? How many cows would it see? Does this fit the clues?
If not, imagine there were two cows. How many sheep would each one see? How many cows would each one see? Does this fit the clues?
How about three cows ...?
Possible extension
Invite children to create some similar questions/situations of their own. They could include more than two different kinds of animal.
Possible support
Having objects available to represent the animals would be a good idea (for example lots of multilink cubes in two different colours).